Question

a political pollster selects a random sample of 75 registered voters from allegheny county, pennsylvania. the city of pittsburgh is in allegheny county. the pollster is interested in determining if the distribution of political party preference among registered voters in this county differs between those who live in pittsburgh and those who live in the suburbs. the responses are displayed in the table.

the pollster would like to know if these data provide convincing evidence of a relationship between residential status and political party in the population of all registered voters in allegheny county. the random and 10% conditions are met, and this table of expected counts shows that all expected counts are at least 5.

what is the value of the chi - square test statistic for this test?

Explanation:

Step1: Recall the chi - square test statistic formula

The formula for the chi - square test statistic is \(\chi^{2}=\sum\frac{(O - E)^{2}}{E}\), where \(O\) is the observed count and \(E\) is the expected count.

Step2: Calculate \(\frac{(O - E)^{2}}{E}\) for each cell

• For the cell (City, Democratic): \(O = 20\), \(E=14.56\). Then \(\frac{(20 - 14.56)^{2}}{14.56}=\frac{5.44^{2}}{14.56}=\frac{29.5936}{14.56}\approx2.033\)
• For the cell (City, Independent): \(O = 8\), \(E = 9.36\). Then \(\frac{(8 - 9.36)^{2}}{9.36}=\frac{(- 1.36)^{2}}{9.36}=\frac{1.8496}{9.36}\approx0.198\)
• For the cell (City, Republican): \(O = 11\), \(E = 15.08\). Then \(\frac{(11 - 15.08)^{2}}{15.08}=\frac{(-4.08)^{2}}{15.08}=\frac{16.6464}{15.08}\approx1.104\)
• For the cell (Suburbs, Democratic): \(O = 8\), \(E = 13.44\). Then \(\frac{(8 - 13.44)^{2}}{13.44}=\frac{(-5.44)^{2}}{13.44}=\frac{29.5936}{13.44}\approx2.202\)
• For the cell (Suburbs, Independent): \(O = 10\), \(E = 8.64\). Then \(\frac{(10 - 8.64)^{2}}{8.64}=\frac{1.36^{2}}{8.64}=\frac{1.8496}{8.64}\approx0.214\)
• For the cell (Suburbs, Republican): \(O = 18\), \(E = 13.92\). Then \(\frac{(18 - 13.92)^{2}}{13.92}=\frac{4.08^{2}}{13.92}=\frac{16.6464}{13.92}\approx1.196\)

Step3: Sum up the values

\(\chi^{2}=2.033 + 0.198+1.104 + 2.202+0.214+1.196\)
\(\chi^{2}=7.0\)

Answer:

\(7.0\)